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In contrast to the Berry connection, which is physical only after integrating around a closed path, the Berry curvature is a gauge-invariant local manifestation of the geometric properties of the wavefunctions in the parameter space, and has proven to be an essential physical ingredient for understanding a variety of electronic properties.
Consider a generic (possibly non-Abelian) gauge transformation acting on a component field = =.The main examples in field theory have a compact gauge group and we write the symmetry operator as () = where () is an element of the Lie algebra associated with the Lie group of symmetry transformations, and can be expressed in terms of the hermitian generators of the Lie algebra (i.e. up to a ...
Various topologically ordered states have interesting properties, such as (1) ground state degeneracy [3] and fractional statistics or non-abelian group statistics that can be used to realize a topological quantum computer; (2) perfect conducting edge states that may have important device applications; (3) emergent gauge field and Fermi ...
In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there exists at least one pair of elements a and b of G, such that a ∗ b ≠ b ∗ a. [1] [2] This class of groups contrasts with the abelian groups, where all pairs of group elements commute.
Non-abelian or nonabelian may refer to: Non-abelian group, in mathematics, a group that is not abelian (commutative) Non-abelian gauge theory, in physics, a gauge ...
In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. [1] Since locally compact abelian groups have a well-understood theory, Pontryagin duality, which includes the basic structures of Fourier series and Fourier transforms, the major business of non-commutative harmonic analysis is ...
The free abelian group on S can be explicitly identified as the free group F(S) modulo the subgroup generated by its commutators, [F(S), F(S)], i.e. its abelianisation. In other words, the free abelian group on S is the set of words that are distinguished only up to the order of letters. The rank of a free group can therefore also be defined as ...
Toën, B. "Stacks and non-abelian cohomology" (PDF). Archived from the original (PDF) on January 14, 2014. Lurie, Jacob (2009). Higher Topos Theory. Annals of ...